A263695 Even numbers such that the sum of the even divisors and the sum of the odd divisors are a square or a cube.

6, 14, 434, 636, 748, 762, 4620, 5964, 6204, 6324, 6580, 6820, 7084, 7660, 8404, 8636, 8804, 9010, 9710, 11342, 11920, 23622, 29820, 31020, 31620, 32844, 35420, 36204, 38964, 39804, 40044, 42020, 43180, 44020, 45724, 46004, 47564, 48484, 49146, 50644, 53444

Offset: 1

Author

Michel Lagneau, May 28 2016

Keywords

• nonn

Comments

It seems that the two sums are never both a square or a cube.

Conjecture [False!]: All squares belonging to a pair are associated with a unique cube. Conversely, all cubes are associated with a unique square.

The corresponding pairs (sum of even divisors, sum of odd divisors) are (2^3, 2^2), (4^2, 2^3), (8^3, 16^2), (36^2, 6^3), (36^2, 6^3), (32^2, 8^3), 11 times the pair (24^3, 48^2), 3 times the pair (108^2, 18^3), (30^3, 30^2), (32^3, 128^2), 16 times the pair (288^2, 24^3),...

We observe several classes of numbers that generate identical pairs, for example:

{636, 748} => pair (36^2, 6^3);

{4620, 5964, 6204, 6324,... } => pair (24^3, 48^2);

{9010, 9710, 11342} => pair (108^2, 18^3);

{29820, 31020, 31620, 32844, 35420,... } => pair (288^2, 24^3);

{69576, 72168, 87752, 98552,...} => pair (56^3, 112^2);

The conjecture above is false. Consider for example the triples of numbers {69576, 938184, 7505472} or {958528, 952520, 12382760}. For the first one the (even, odd) sum of divisors pairs are (56^3, 112^2), (1568^2, 56^3), and (4704^2, 56^3). - Giovanni Resta, May 28 2016

Examples

			434 is in the sequence because the divisors are {1, 2, 7, 14, 31, 62, 217, 434} => sum of even divisors = 2+14+62+434 = 512 = 8^3 and sum of odd divisors = 1+7+31+217 = 256 = 16^2.
636 is in the sequence because the divisors are {1, 2, 3, 4, 6, 12, 53, 106, 159, 212, 318, 636} => sum of even divisors = 2+4+6+12+106+212+318+636 = 1296 = 36^2 and sum of odd divisors = 1+3+53+159 = 216 = 6^3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A074400, A146076.

Programs

Maple

with(numtheory):
for n from 2 by 2  to 500000 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     ii:=0:
        for a from 1 to 1000 while(ii=0)do:
        for i from 2 to 3 do:
         if s0=a^i
          then
           for b from 1 to 1000 while(ii=0) do:
             if s1=b^(5-i)
              then
              ii:=1:printf(`%d, `,n):
              else
             fi:
           od:
          fi:
        od:
      od:
     od:

Mathematica

es[n_] := 2 DivisorSigma[1, n/2]; os[n_] := DivisorSigma[1, n] - es[n]; powQ[n_] := Or @@ IntegerQ /@ (n^(1/{2, 3})); Select[2 Range[10^4], powQ@ es@ # && powQ@ os@ # &] (* Giovanni Resta, May 28 2016 *)

PARI

isA002760(n)=issquare(n) || ispower(n,3)
is(n)=n%2==0 && isA002760(2*sigma(n/2)) && isA002760(sigma(n>>valuation(n,2))) \\ Charles R Greathouse IV, Jun 08 2016